Model reduction for a power grid model

Jing Li; Panos Stinis

doi:10.3934/jcd.2021019

Article Contents

2022, Volume 9, Issue 1: 1-26. Doi: 10.3934/jcd.2021019

This issue Previous Article Next Article Motion tomography via occupation kernels

Model reduction for a power grid model

Jing Li and
Panos Stinis^,

Pacific Northwest National Laboratory, Richland, WA 99354, USA

^* Corresponding author: Panos Stinis
^* Corresponding author: Panos Stinis

Received: November 2020

Revised: November 2021

Early access: December 15, 2021

Published online: December 15, 2021

The work presented here was supported by the U.S. Department of Energy (DOE) Office of Science, Office of Advanced Scientific Computing Research (ASCR) as part of the Multifaceted Mathematics for Rare, Extreme Events in Complex Energy and Environment Systems (MACSER) project. Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.

Abstract / Introduction Full Text(HTML) Figure(11) / Table(1) Related Papers Cited by

Abstract

We examine the complexity of constructing reduced order models for subsets of the variables needed to represent the state of the power grid. In particular, we apply model reduction techniques to the DeMarco-Zheng power grid model. We show that due to the oscillating nature of the solutions and the absence of timescale separation between resolved and unresolved variables, the construction of accurate reduced models becomes highly non-trivial because one has to account for long memory effects. In addition, we show that a reduced model that includes even a short memory is drastically better than a memoryless model.

Keywords:

Mathematics Subject Classification: Primary: 65L99; Secondary: 65Z05.

Citation:

Full Text(HTML)

Figure 1. A 3-bus system.

Download: Full-size image PowerPoint slide

Figure 2. Full system - Evolution of resolved variables. a) $ \omega_1 $ for generator 1, b) $ \omega_2 $ for generator 2 and c) $ \alpha_2 $ for generator 2.

Download: Full-size image PowerPoint slide

Figure 3. Full system - Evolution of unresolved variables. a) $ \alpha_3 $ for load bus 3, b) $ V_3 $ for load bus 3.

Download: Full-size image PowerPoint slide

Figure 4. Reduced system - Evolution of the memory integral. (a) For $ \omega_1, $ (b) For $ \omega_2. $.

Download: Full-size image PowerPoint slide

Figure 5. Reduced model for 3-bus system. Comparison of reduced models with infinite memory and without memory. (a) Evolution of the resolved variable $ \alpha_2, $ (b) Evolution of the resolved variable $ \omega_1 $, and (c) Evolution of the resolved variable $ \omega_2. $

Download: Full-size image PowerPoint slide

Figure 6. Reduced model for 3-bus system. Comparison of reduced models with variable memory length (including without memory). (a) Evolution of the resolved variable $ \alpha_2 $ and (b) Evolution of the resolved variable $ \omega_1. $

Download: Full-size image PowerPoint slide

Figure 7. Reduced model for 3-bus system. Comparison of reduced models with variable order of the finite-rank projection and timestep $ \Delta t = 10^{-4}. $ (a) Evolution of the resolved variable $ \alpha_2, $ (b) Evolution of the resolved variable $ \omega_1. $

Download: Full-size image PowerPoint slide

Figure 8. Reduced model for the 3-bus system. Evolution of the memory kernels $ b_1^{\mu}(s) $ of the resolved variable $ \omega_1 $ on linear functions of the resolved variables. (a) Projection on the 1 degree Hermite polynomial $ h^{(1,0,0)}, $ (b) Projection on the 1 degree Hermite polynomial $ h^{(0,1,0)}, $ and (c) Projection on the 1 degree Hermite polynomial $ h^{(0,0,1)}. $ We have also included in the insets the evolution near the time origin (see text for details).

Download: Full-size image PowerPoint slide

Figure 9. Reduced model for 3-bus system. Evolution of the projections $ f_1^{\mu}(s) = (Le^{sL}F_1(u_0,0),h^{\mu}(\hat{u}_0)) $ of $ Le^{sL}F_1(u_0,0) $ on linear functions of the resolved variables. (a) Projection on the 1 degree Hermite polynomial $ h^{(1,0,0)}, $ (b) Projection on the 1 degree Hermite polynomial $ h^{(0,1,0)}, $ and (c) Projection on the 1 degree Hermite polynomial $ h^{(0,0,1)}. $ We have also included in the insets the evolution near the time origin (see text for details).

Download: Full-size image PowerPoint slide

Figure 10. Reduced model for particle coupled to heat bath - Evolution of the position $ x(t) $ of the particle (a) For $ t_{memory} = 1, $ (b) For $ t_{memory} = 2, $ and (c) For $ t_{memory} = 3. $

Download: Full-size image PowerPoint slide

Figure 11. Reduced model for particle coupled to heat bath - Evolution of the momentum $ p(t) $ of the particle (a) For $ t_{memory} = 1, $ (b) For $ t_{memory} = 2, $ and (c) For $ t_{memory} = 3. $

Download: Full-size image PowerPoint slide

Table 1. Parameters for the 3-bus system

$ M_1 $	$ M_2 $	$ b_1 $	$ b_2 $	$ b_3 $	$ D_1 $	$ D_2 $
$ 0.052 $	$ 0.0531 $	$ 10 $	$ 10 $	$ 10 $	$ 0.05 $	$ 0.05 $
$ D_3 $	$ P_2 $	$ P_3 $	$ Q_3 $	$ \epsilon $	$ V_1 $	$ V_2 $
$ 0.005 $	$ -2.0 $	$ 3.0 $	$ 0.1 $	$ 5 $	$ 0.9 $	$ 0.9 $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	M. Ceriotti, G. Bussi and M. Parrinello, Colored-noise thermostats à la carte, J. Chem. Theory Comput., 6 (2010), 1170-1180. doi: 10.1021/ct900563s.
[2]	A. J. Chorin, O. H. Hald and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968-2973. doi: 10.1073/pnas.97.7.2968.
[3]	A. J. Chorin, O. H. Hald and R. Kupferman, Optimal prediction with memory, Phys. D, 166 (2002), 239-257. doi: 10.1016/S0167-2789(02)00446-3.
[4]	A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Commun. Appl. Math. Comput. Sci., 1 (2006), 1-27. doi: 10.2140/camcos.2006.1.1.
[5]	J. H. Chow, Power System Coherency and Model Reduction, Power Electronics and Power Systems, 94, Springer, New York, 2013. doi: 10.1007/978-1-4614-1803-0.
[6]	I. Dobson, B. A. Carreras, V. E. Lynch and D. E. Newman, Newman, Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization, Chaos, 17 (2007), 1-27. doi: 10.1063/1.2737822.
[7]	D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), R55-R127. doi: 10.1088/0951-7715/17/6/R01.
[8]	O. H. Hald and P. Stinis, Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions, Proc. Natl. Acad. Sci. USA, 104 (2007), 6527-6532. doi: 10.1073/pnas.0700084104.
[9]	H. S. Lee, S.-H. Ahn and E. F. Darve, The multi-dimensional generalized Langevin equation for conformational motion of proteins, J. Chem. Phys., 150 (2019). doi: 10.1063/1.5055573.
[10]	J. Li and P. Stinis, A unified framework for mesh refinement in random and physical space, J. Comput. Phys., 323 (2016), 243-264. doi: 10.1016/j.jcp.2016.07.027.
[11]	J. Li and P. Stinis, Mori-Zwanzig reduced models for uncertainty quantification, J. Comput. Dyn., 6 (2019), 39-68. doi: 10.3934/jcd.2019002.
[12]	A. E. Motter, S. A. Myers, M. Anghel and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nature Physics, 9 (2013), 191-197. doi: 10.1038/nphys2535.
[13]	D. Osipov and K. Sun, Adaptive nonlinear model reduction for fast power system simulation, IEEE Trans. Power Syst., 33 (2018), 6746-6754.
[14]	J. Qi, J. Wang, H. Liu and A. D. Dimitrovski, Nonlinear model reduction in power systems by balancing of empirical controllability and observability covariances, IEEE Trans. Power Syst., 32 (2017), 114-126. doi: 10.1109/TPWRS.2016.2557760.
[15]	I. Simonsen, K. Buzna and S. Peters, Bornholdt and D. Helbing, Transient dynamics increasing network vulnerability to cascading failures, Phys. Rev. Lett., 100 (2008). doi: 10.1103/PhysRevLett.100.218701.
[16]	F. Sloothaak, S. C. Borst and B. Zwart, Robustness of power-law behavior in cascading line failure models, Stoch. Models, 34 (2018), 45-72. doi: 10.1080/15326349.2017.1383165.
[17]	P. Stinis, A phase transition approach to detecting singularities of partial differential equations, Commun. Appl. Math. Comput. Sci., 4 (2009), 217-239. doi: 10.2140/camcos.2009.4.217.
[18]	P. Stinis, Renormalized Mori-Zwanzig-reduced models for systems without scale separation, Proc. A., (2015), 13pp. doi: 10.1098/rspa.2014.0446.
[19]	S. Tamrakar, M. Conrath and S. Kettemann, Propagation of disturbances in AC electricity grids, Scientific Reports, 8 (2018). doi: 10.1038/s41598-018-24685-5.
[20]	D. Witthaut and M. Timme, Braess's paradox in oscillator networks, desynchronization and power outage, New J. Phys., 14 (2012). doi: 10.1088/1367-2630/14/8/083036.
[21]	D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139. doi: 10.1137/040615201.
[22]	H. Zheng and C. L. DeMarco, A bi-stable branch model for energy-based cascading failure analysis in power systems, North American Power Symposium 2010, Arlington, TX, 2010. doi: 10.1109/NAPS. 2010.5618968.
[23]	R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001.